November 18, 2024
A popular regression modeling method used to explore the relationship between survival time and covariates.
It assumes that the effects of different variables on the outcome are constant over time.
Survival can refer to the development of a symptom, time to relapse after remission, or as a time to death [1].
Cox regression model is based on the hazard function \(h_x(t)\) with covariates at time t given by [2]:
\(h_x(t)=h_0(t)\exp(\beta_1x_1 +\beta_2x_2 + \dots + \beta_p x_p)\)
Where:
\(h_x(t)\) is the hazard function
\(h_0(t)\) is the baseline hazard function
\(\beta_1x_1 + \beta_2x_2 + \dots +\beta_p x_p\) represent the linear combination of covariates and their coefficient
The hazard function is the probability that an individual will experience an event (death) within a certain time interval [1].
The hazard ratio is used to compare the hazard rate between two groups
HR = \(hx_2(t)\) / \(hx_1(t)\) = \(\exp[\beta(x_2-x_1)]\)
HR = 1: No difference in hazard rates between the two groups
HR >1: Higher hazard rate in the second group compared to the first
HR <1: Lower hazard rate in the second group compared to the first
Failing to meet the assumption of proportional hazards means that the effects between dependent and explanatory variables are not constant over time.
Time-varying covariates (coefficients) are used when a covariate changes over time during the follow-up period [4].
Internal time-varying coefficients are affected by survival status and include values that are generated by the subject [4].
External time-varying coefficients are pre-determined and external to the subject under study [4].
| Package | Description |
|---|---|
| Survival | - Used for fitting and analyzing survival models - Fits Kaplan-Meier survival curves |
| Survminer | - Plots Kaplan-Meier survival curves using ggplot2 - Plots Schoenfeld residuals |
Data: Survival after chemotherapy for Stage B/C colon cancer [6]
Goal: Model the relationship between survival time and treatment groups
Predictors
| Category | Variables |
|---|---|
| Treatments | - Observation (no treatment) - Amisole (Lev) - Amisole + 5-FU |
| Patient Characteristics | - Age - Sex |
| Tumor Characteristics | - Colon perforation and obstruction - Adherence to nearby organs - Tumor differentiation - Local spread |
Base model: No predictors
Univariate: Treatment
Full variables: All predictors
Significant predictors: stepwise-selected variables
Final model: Stratified
| Characteristic | HR1 | 95% CI1 | p-value |
|---|---|---|---|
| rx | |||
| Obs | — | — | |
| Lev | 0.97 | 0.78, 1.21 | 0.8 |
| Lev+5FU | 0.69 | 0.55, 0.87 | 0.002 |
| Concordance = 0.536 | |||
| 1 HR = Hazard Ratio, CI = Confidence Interval | |||
| Characteristic | HR1 | 95% CI1 | p-value |
|---|---|---|---|
| rx | |||
| Obs | — | — | |
| Lev | 0.98 | 0.79, 1.22 | 0.9 |
| Lev+5FU | 0.69 | 0.54, 0.87 | 0.002 |
| age | 1.01 | 1.00, 1.02 | 0.083 |
| sex | 1.04 | 0.86, 1.26 | 0.7 |
| perfor | 1.00 | 0.59, 1.70 | >0.9 |
| adhere | 1.18 | 0.92, 1.53 | 0.2 |
| surg | 1.27 | 1.03, 1.55 | 0.022 |
| obstruct | 1.33 | 1.06, 1.68 | 0.015 |
| differentiation | |||
| moderate | — | — | |
| poor | 1.43 | 1.13, 1.82 | 0.003 |
| well | 1.08 | 0.78, 1.50 | 0.6 |
| node4 | 2.55 | 2.10, 3.09 | <0.001 |
| local_spread | |||
| contiguous | — | — | |
| muscle | 0.39 | 0.23, 0.64 | <0.001 |
| serosa | 0.64 | 0.43, 0.94 | 0.023 |
| submucosa | 0.29 | 0.10, 0.83 | 0.021 |
| Concordance = 0.674 | |||
| 1 HR = Hazard Ratio, CI = Confidence Interval | |||
| Characteristic | HR1 | 95% CI1 | p-value |
|---|---|---|---|
| rx | |||
| Obs | — | — | |
| Lev | 0.99 | 0.80, 1.23 | >0.9 |
| Lev+5FU | 0.69 | 0.54, 0.87 | 0.002 |
| age | 1.01 | 1.00, 1.02 | 0.069 |
| surg | 1.28 | 1.04, 1.56 | 0.018 |
| obstruct | 1.33 | 1.06, 1.67 | 0.015 |
| differentiation | |||
| moderate | — | — | |
| poor | 1.45 | 1.15, 1.84 | 0.002 |
| well | 1.07 | 0.77, 1.48 | 0.7 |
| node4 | 2.53 | 2.09, 3.07 | <0.001 |
| local_spread | |||
| contiguous | — | — | |
| muscle | 0.37 | 0.23, 0.61 | <0.001 |
| serosa | 0.61 | 0.41, 0.89 | 0.010 |
| submucosa | 0.27 | 0.09, 0.76 | 0.014 |
| Concordance = 0.672 | |||
| 1 HR = Hazard Ratio, CI = Confidence Interval | |||
| chisq | df | p | Variable | |
|---|---|---|---|---|
| rx | 2.335 | 2 | 0.311 | rx |
| age | 0.549 | 1 | 0.459 | age |
| surg | 0.020 | 1 | 0.888 | surg |
| obstruct | 6.148 | 1 | 0.013 | obstruct |
| differentiation | 17.459 | 2 | 0.000 | differentiation |
| node4 | 5.662 | 1 | 0.017 | node4 |
| local_spread | 7.083 | 3 | 0.069 | local_spread |
| GLOBAL | 37.525 | 11 | 0.000 | GLOBAL |
| Characteristic | HR1 | 95% CI1 | p-value |
|---|---|---|---|
| rx | |||
| Obs | — | — | |
| Lev | 0.98 | 0.79, 1.22 | 0.9 |
| Lev+5FU | 0.71 | 0.56, 0.89 | 0.003 |
| age | 1.01 | 1.00, 1.02 | 0.034 |
| surg | 1.30 | 1.06, 1.59 | 0.012 |
| node4 | 2.50 | 2.06, 3.04 | <0.001 |
| local_spread | |||
| contiguous | — | — | |
| muscle | 0.34 | 0.21, 0.56 | <0.001 |
| serosa | 0.58 | 0.39, 0.84 | 0.004 |
| submucosa | 0.24 | 0.08, 0.67 | 0.007 |
| Concordance = 0.674 | |||
| 1 HR = Hazard Ratio, CI = Confidence Interval | |||
| chisq | df | p | Variable | |
|---|---|---|---|---|
| rx | 2.001 | 2 | 0.368 | rx |
| age | 0.670 | 1 | 0.413 | age |
| surg | 0.014 | 1 | 0.905 | surg |
| node4 | 4.288 | 1 | 0.038 | node4 |
| local_spread | 5.298 | 3 | 0.151 | local_spread |
| GLOBAL | 12.411 | 8 | 0.134 | GLOBAL |
| Model | Description | AIC | BIC | C_Index |
|---|---|---|---|---|
| Model 1 | Base model | 5860.383 | 5860.383 | 0.500 |
| Model 2 | Treatment | 5852.236 | 5860.463 | 0.536 |
| Model 3 | Full variables | 5741.401 | 5798.993 | 0.674 |
| Model 4 | Stepwise-selected variables | 5737.261 | 5782.511 | 0.672 |
| Model 5 | Stratified | 4567.829 | 4600.739 | 0.674 |
Original c-index: 0.6544784
Mean cross-validated c-Index: 0.6420104